## Weighted norm inequalities of Hardy type for a class of integral operators.(English)Zbl 0837.26012

Let $$u$$ and $$v$$ be nonnegative measurable functions on $$(0, \infty)$$. Let $Kf(x)= \int^x_0 k(x, y) f(y) dy,$ where the kernel $$k$$ satisfies the following conditions: $k(x, y)\geq 0\quad\text{for} \quad 0< y< x\tag{i}$ and it is non-decreasing in $$x$$ or non-increasing in $$y$$, $D^{- 1}(k(x, y)+ k(y, z))\leq k(x, z)\leq D(k(x, y)+ k(y, z))\quad\text{ if }\quad 0< z< y< x,\tag{ii}$ where $$D\geq 1$$ is independent of $$x$$, $$y$$ and $$z$$.
For the cases in which $$1< p,q< \infty$$ necessary and sufficient conditions on $$u$$ and $$v$$ are found for the following inequality to hold with a constant $$C> 0$$ independent of $$f$$: $|(Kf) u|_q\leq C|fv|_p,\tag{1}$ where $$|\cdot|_s$$ is the $$L_s$$- norm on $$(0, \infty)$$. Necessary and sufficient conditions of compactness of the operator $$K$$ are obtained as well. If $$0< q< 1< p< \infty$$ then necessary conditions and sufficient conditions for validity of (1) are also obtained.

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 42B99 Harmonic analysis in several variables
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